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From Static to Dynamic Tag Population Estimation: An Extended Kalman Filter Perspective Jihong Yu, Lin Chen, Rongrong Zhang, Kehao Wang

Abstract—Tag population estimation has recently attracted significant research attention due to its paramount importance on a variety of radio frequency identification (RFID) applications. However, most, if not all, of existing estimation mechanisms are proposed for the static case where tag population remains constant during the estimation process, thus leaving the more challenging dynamic case unaddressed, despite the fundamental importance of the latter case on both theoretical analysis and practical application. In order to bridge this gap, we devote this paper to designing a generic framework of stable and accurate tag population estimation schemes based on Kalman filter for both static and dynamic RFID systems. Technically, we first model the dynamics of RFID systems as discrete stochastic processes and leverage the techniques in extended Kalman filter (EKF) and cumulative sum control chart (CUSUM) to estimate tag population for both static and dynamic systems. By employing Lyapunov drift analysis, we mathematically characterise the performance of the proposed framework in terms of estimation accuracy and convergence speed by deriving the closed-form conditions on the design parameters under which our scheme can stabilise around the real population size with bounded relative estimation error that tends to zero with exponential convergence rate. Index Terms—RFID, tag population estimation, extended Kalman filtre, stochastic stability.

I. I NTRODUCTION A. Context and Motivation Recent years have witnessed an unprecedented development and application of the radio frequency identification (RFID) technology. As a promising low-cost technology, RFID is widely utilized in various applications ranging from inventory control [1][2], supply chain management [3] to tracking/location [4][5]. A standard RFID system has two types of devices: a set of RFID tags and one or multiple RFID readers (simply called tags and readers). A tag is typically a lowcost microchip labeled with a unique serial number (ID) to identify an object. A reader, on the other hand, is equipped with an antenna and can collect the information of tags within its coverage area. Tag population estimation and counting is a fundamental functionality for many RFID applications such as warehouse management, inventory control and tag identification. For example, quickly and accurately estimating the number of tagged objects is crucial in establishing inventory reports for large retailers such as Wal-Mart [6]. Due to the paramount J. Yu and L. Chen are with Lab. Recherche Informatique (LRI-CNRS UMR 8623), Univ. Paris-Sud, 91405 Orsay, France, {Jihong.Yu, Lin.Chen}@lri.fr. R. Zhang is with Lab. Informatique, Univ. Paris Descartes, France, [email protected]. K. Wang is with Dept. Inform. Eng., Wuhan University of Technology, China, [email protected].

practical importance of tag population estimation, a large body of studies [7][8][9][10][11] have been devoted to the design of efficient estimation algorithms. Most of them, as reviewed in Sec. II, are focused on the static scenario where the tag population is constant during the estimation process. However, many practical RFID applications, such as logistic control, are dynamic in the sense that tags may be activated or terminated as specialized in C1G2 standard [12], or the tagged objects may enter and/or leave the reader’s covered area frequently, thus resulting in tag population variation. In such dynamic applications, a fundamental research question is how to design efficient algorithms to dynamically trace the tag population quickly and accurately. B. Summary of Contributions In this paper, we develop a generic framework of stable and accurate tag population estimation schemes for both static and dynamic RFID systems. By generic, we mean that our framework both supports the real-time monitoring and can estimate the number of tags accurately without any prior knowledge on the tag arrival and departure patterns. Our design is based on the extended Kalman filter (EKF) [13], a powerful tool in optimal estimation and system control. By performing Lyapunov drift analysis, we mathematically prove the efficiency and stability of our framework in terms of the boundedness of estimation error. The main technical contributions of this paper are articulated as follows. We formulate the system dynamics of the tag population for both static and dynamic RFID systems where the number of tags remains constant and varies during the estimation process. We design an EKF-based population estimation algorithm for static RFID systems and further enhance it to dynamic RFID systems by leveraging the cumulative sum control chart (CUSUM) to detect the population change. By employing Lyapunov drift analysis, we mathematically characterise the performance of the proposed framework in terms of estimation accuracy and convergence speed by deriving the closed-form conditions on the design parameters under which our scheme can stabilise around the real population size with bounded relative estimation error that tends to zero within exponential convergence rate. To the best of our knowledge, our work is the first theoretical framework that dynamically traces the tag population with closed form conditions on the estimation stability and accuracy. II. R ELATED W ORK Due to its fundamental importance, tag population estimation has received significant research attention, which we

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briefly review in this section. A. Tag Population Estimation for Static RFID systems Most of existing works are focused on the static scenario where the tag population is constant during the estimation process. The central question there is to design efficient algorithms quickly and accurately estimating the static tag population. Kodialam et al. design an estimator called PZE which uses the probabilistic properties of empty and collision slots to estimate the tag population size [14]. The authors then further enhance PZE by taking the average of the probability of idle slots in multiple frames as an estimator in order to eliminate the constant additive bias [7]. Han et al. exploit the average number of idle slots before the first non-empty slots to estimate the tag population size [15]. Later, Qian et al. develop Lottery-Frame scheme that employs geometrically distributed hash function such that the jth slot is chosen with 1 [9]. As a result, the first idle slot approaches prob. 2j+1 around the logarithm of the tag population and the frame size can be reduced to the logarithm of the tag population, thus reducing the estimation time. Subsequently, a new estimation scheme called ART is proposed in [10] based on the average length of consecutive non-empty slots. The design rational of ART is that the average length of consecutive non-empty slots is correlated to the tag population. ART is shown to have smaller variance than prior schemes. More recently, Zheng et al. propose another estimation algorithm, ZOE, where each frame just has a single slot and the random variable indicating whether a slot is idle follows Bernoulli distribution [11]. The average of multiple individual observations is used to estimate the tag population. We would like to point out that the above research work does not consider the estimation problem for dynamic RFID systems and thus may fail to monitor the system dynamics in real time. Specifically, in typical static tag population estimation schemes, the final estimation result is the average of the outputs of multi-round executions. When applied to dynamic tag population estimation, additional estimation error occurs due to the variation of the tag population size during the estimation process. B. Tag Population Estimation for Dynamic RFID systems Only a few propositions have tackled the dynamic scenario. The works in [16] and [17] consider specific tag mobility patterns that the tags move along the conveyor in a constant speed, while tags may move in and out by different workers from different positions, so these two algorithm cannot be applicable to generic dynamic scenarios. Subsequently, Xiao et al. develop a differential estimation algorithm, ZDE, in dynamic RFID systems to estimate the number of arriving and removed tags [18]. More recently, they further generalize ZDE by taking into account the snapshots of variable frame sizes [19]. Though the algorithms in [18] and [19] can monitor the dynamic RFID systems, they may fail to estimate the tag population size accurately, because they must use the same hash seed in the whole monitoring process, which cannot reduce the estimation variance. Using the same seed is an

effective way in tracing tag departure and arrival. However, it may significantly limit the estimation accuracy, even in the static case. Besides the limitations above, prior works do not provide formal analysis on the stability and the convergence rate. To fill this vide, we develop a generic framework for tag population estimation in dynamic RFID systems. By generic, we mean that our framework can both support real-time monitoring and estimate the number of tags accurately without the requirement for any prior knowledge on the tag arrival and departure patterns. As another distinguished feature, the efficiency and stability of our framework is mathematically established. III. T ECHNICAL P RELIMINARIES In this section, we briefly introduce the extended Kalman filter and some fundamental concepts and results in stochastic process which are useful in the subsequent analysis. A. Extended Kalman Filter The extended Kalman filter is a powerful tool to estimate system state in nonlinear discrete-time systems. Formally, a nonlinear discrete-time system can be described as follows: zk+1 = f (zk , xk ) + wk∗ (1) ∗ yk = h(zk ) + uk , (2) where zk+1 ∈ Rn denotes the state of the system, xk ∈ Rd is the controlled inputs and yk ∈ Rm stands for the measurement observed from the system. The uncorrelated stochastic variables wk∗ ∈ Rn and u∗k ∈ Rm denote the process noise and the measurement noise, respectively. The functions f and h are assumed to be the continuously differentiable. For the above system, we introduce an EKF-based state estimator given in Definition 1. Definition 1 (Extended Kalman filter [13]). A two-step discrete-time extended Kalman filter consists of state prediction and measurement update, defined as 1) Time update (prediction) zˆk+1|k = f (ˆ zk|k , xk ) (3) Pk+1|k = Pk|k + Qk , (4) 2) Measurement update (correction) zˆk+1|k+1 = f (ˆ zk+1|k , xk ) + Kk+1 vk+1 (5) Pk+1|k+1 = Pk+1|k (1 − Kk+1 Ck+1 ) (6) Pk+1|k Ck+1 , (7) Kk+1 = Pk+1|k Ck+1 2 + Rk+1 where vk+1 = yk+1 − h(ˆ z ) k+1|k ∂h(zk+1 ) Ck+1 = . ∂zk+1 zk+1 =ˆzk+1|k

(8) (9)

Remark. In the above definition of extended Kalman filter, the parameters can be interpreted in our context as follows: • z ˆk+1|k is the prediction of zk+1 at the beginning of frame k +1 given by the previous state estimate, while zˆk+1|k+1 is the estimate of zk+1 after the adjustment based on the measure at the end of frame k + 1.



•

• •

•

vk+1 , referred to as innovation, is the measurement residual in frame k+1. It represents the estimated error of the measure. Kk+1 is the Kalman gain. With reference to equation (5), it weighs the innovation vk+1 w.r.t. f (ˆ zk+1|k , xk ). Pk+1|k and Pk+1|k+1 , in contrast to the linear case, are not equal to the covariance of estimation error of the system state. Here, we will refer to them as pseudocovariance. Qk and Rk are two tunable parameters which play the role as that of the covariance of the process and measurement noises in linear stochastic systems to achieve optimal filtering in the maximum likelihood sense. We will show later that Qk and Rk also play an important role in improving the stability and convergence of our EKFbased estimators.

i.e, (14) and (13) hold, then ek|k−1 is also bounded and the convergence rate depends on constant α mostly. Besides, it can be noted that Lemma 1 can only be implemented offline. To address this limit, we adjust Lemma 1 to an online version with time-varying parameters, which can be proven by the same method as in [21] and [23]. Lemma 2. If there exist a stochastic process Vk (ek|k−1 ) and real numbers β ∗ , βk , τk >0 and 0 0 such that lim sup P{|ek|k−1 | > X} = 0. (11) k→∞ k≥1

Definition 3 (Boundedness in Mean Square). The stochastic process ek|k−1 is said to be exponentially bounded in the mean square with exponent ζ, if there exist real numbers ψ1 , ψ2 > 0 and 0 < ζ < 1 such that E[e2k|k−1 ] ≤ ψ1 e21|0 ζ k−1 + ψ2 . (12) To investigate the boundedness defined in Definition 2 and 3, we introduce the following lemma [22]. Lemma 1. Given a stochastic process Vk (ek|k−1 ) and constants β, β, τ >0 and 0 0, there exists some M > 0, such that if zk ≥ M or Lk = zˆk|k−1 ≥ M , then it holds that µ − Lk e−rk ρ ≤ ∗ , (21) 2 σ − Lk (e−rk ρ − (1 + rk2 ρ)e−2rk ρ ) ≤ ∗ , (22) where ρ = Lzkk is referred to as the reader load factor.

Our objective is to design a stable and accurate tag population estimation algorithm for both static and dynamic systems. By stable and accurate we mean that • the estimation error of our algorithm is bounded in the sense of Definition 2 and 3 and the relative estimation error tends to zero; • the estimated population size converges to the real value with exponential rate. Mathematically, we consider a large-scale RFID system of a reader and a set of tags with the unknown size zk in frame k which can be static or dynamic. Denote by zˆk|k−1 the prior estimate of zk in the beginning of frame k. At the end of frame k, the reader updates the estimate zˆk|k−1 to zˆk|k by running the estimation algorithm. Our designed estimation scheme need to guarantee the following properties: zˆk|k−1 − zk = 0; • lim zk →∞ zk • the converges rate is exponential.

Lemmas 3 and 4 imply that in large-scale RFID systems, we can use Lk e−rk ρ and Lk (e−rk ρ − (1 + rk2 ρ)e−2rk ρ ) to approximate µ and σ 2 . At the end of each frame k, the reader gets a measure yk of the idle slot frequency defined as Nk yk = . (23) Lk Recall Lemma 3, it holds that yk is a Normal distributed random variable specified as follows: E[yk ] = e−rk ρ and V ar[yk ] = L1k (e−rk ρ − (1 + rk2 ρ)e−2rk ρ ). Since there are zk tags reply in frame k with probability rk , the probability that a slot is idle, denoted as p(zk ), can be calculated as r z rk zk − k k (24) p(zk ) = (1 − ) ≈ e Lk . Lk Notice that for large zk , p(zk ) can be regarded as a continuously differentiable function of zk . Using the language in the Kalman filter, we can write yk as follows: yk = p(zk ) + uk , (25)

V. TAG P OPULATION E STIMATION : S TATIC S YSTEMS In this section, we focus on the baseline scenario of static systems where the tag population is constant during the estimation process. We first establish the discrete-time model for the system dynamics and the measurement model using the bit string Bk observed during frame k. We then present our EKF-based estimation algorithm.

where, based on the statistic characteristics of yk , uk is a Gaussian random variable with zero mean and variance 1 −rk ρ V ar[uk ] = (e − (1 + rk2 ρ)e−2rk ρ ). (26) Lk We note that uk measures the uncertainty of yk . To summarise, the discrete-time model for static RFID systems is characterized by (20) and (25). B. Tag Population Estimation Algorithm

A. System Dynamics and Measurement Model Consider the static RFID systems where the tag population stays constant, the system state evolves as zk+1 = zk , (20) meaning that the number of tags zk+1 in the system in frame k + 1 equals that in frame k. In order to estimate zk , we leverage the measurement on the number of idle slots during a frame. To start, we study the stochastic characteristics of the number of idle slots. Assume that the initial tag population z0 falls in the interval z0 ∈ [z 0 , z 0 ], yet the exact value of z0 is unknown and should be estimated. The range [z 0 , z 0 ] can be a very coarse estimation that can be obtained by any existing population estimation method. Recall the system model that in frame k, the reader probes the tags with the frame size Lk . Denote by variable Nk the number of idle slots in frame k, that is, the number of ‘0’s in Bk , we have the following results on Nk according to [14], [25].

Noticing that the system state characterised by (20) and (25) is a discrete-time nonlinear system, we thus leverage the twostep EKF described in Definition 1 to estimate the system state. In (7), the Kalman gain Kk increases with Qk while decreases with Rk . As a result, Qk and Rk can be used to tune the EKF such that increasing Qk and/or decreasing Rk accelerates the convergence rate but leads to larger estimation error. In our design, we set Qk to a constant q > 0 and introduce a parameter φk as follows to replace Rk to facilitate our demonstration: Rk = φk Pk|k−1 Ck 2 . (27) It can be noted from (7) and (27) that Kk is monotonously decreasing in φk , i.e., a small φk leads to quick convergence with the price of relatively high estimation error. Hence, choosing the appropriate value for φk consists of striking a balance between the convergence rate and the estimation error. In our work, we take a dynamic approach by setting φk to a small value φ but satisfying (62) at the first few rounds (J rounds) of estimation to allow the system to act quickly since



the estimation in the beginning phase can be very coarse. After that we set φk to a relatively high value φ to achieve high estimation accuracy. Now, we present our tag population estimation algorithm in Algorithm 1 where P0|0 , q can be set to some constants straightforward since the performance mostly depends on φk and kmax is the time horizon during which the system needs to be monitored, e.g., if the RFID system needs to be monitored from time T1 to T2 , then at T1 , kmax is simply set as T2 − T1 . The major procedures can be summarised as: 1) In the beginning of frame k: prediction (line 3). The reader first predicts the number of tags based on the estimation at the end of frame k−1. The predicted value is defined as zˆk|k−1 . Then the reader sets the persistence probability rk following Lemma 8 and zk is set to zˆk|k−1 . 2) Line 4-5. The reader launches the Listen-before-talk protocol as introduced in IV-A in order to receive the feedbacks from tags. 3) At the end of frame k: correction (line 6-14). The reader computes Nk based on Bk and further calculates yk and vk from Nk . It then updates the prediction with the corrected estimate zˆk|k following (5). We will theoretically establish the stability and accuracy of the algorithm in Sec. VII. Algorithm 1 Tag population estimation (static cases): executed by the reader Input: z 0 , P0|0 , q, J, L, φ, φ, maximum number of rounds kmax Output: Estimated tag population set Sz = {ˆ zk|k : k ∈ [0, kmax ]} 1: Initialisation: z ˆ0|0 ← z 0 , Q0 ← q, Sz = {ˆ z0|0 } 2: for k = 1 to kmax do 3: zˆk|k−1 ← zˆk−1|k−1 , Lk ← L, rk ← 1.59Lk /ˆ zk|k−1 , Pk|k−1 ← Pk−1|k−1 + Qk−1 4: Generate a new random seed Rsk and broadcast (Lk , rk , Rsk ) 5: Run Listen-before-Talk protocol 6: Obtain the number of idle slots Nk , and compute yk and vk using (23) and (8) 7: Qk ← q 8: if k ≤ J then 9: φk ← φ 10: else 11: φk ← φ 12: end if 13: Calculate Rk and Kk using (27) and (7) 14: Update zˆk|k and Pk|k using (5) and (6) 15: Sz ← Sz ∪ {ˆ zk|k } 16: end for VI. TAG P OPULATION E STIMATION : DYNAMIC S YSTEMS In this section, we further tackle the dynamic case where the tag population may vary during the estimation process. The objective for the dynamic systems is to promptly detect the global tag papulation change and accurately estimate the

quantity of this change. To that end, we first establish the system model and then present our estimation algorithm. A. System Dynamics and Measurement Model In dynamic RFID systems, we can formulate the system dynamics as zk+1 = zk + wk , (28) where the tag population zk+1 in frame k+1 consists of two parts: i) the tag population in frame k and ii) a random variable wk which accounts for the stochastic variation of tag population resulting from the tag arrival/departure during frame k. Notice that wk is referred to as process noise in Kalman filters and the appropriate characterisation of wk is crucial in the design of stable Kalman filters, which will be investigated in detail later. Besides, the measurement model is the same as the static case. Hence, the discrete-time model for dynamic RFID systems can be characterized by (28) and (25). B. Tag Population Estimation Algorithm In the dynamic case, we leverage the two-step EKF to estimate the system state combined with the CUSUM test to further trace the tag population fluctuation. Algorithm 2 Tag population estimation (unified framework): executed by the reader Input: z 0 , P0|0 , q, J, L, φ, φ, maximum number of rounds kmax Output: Estimation set Sz = {ˆ zk|k : k ∈ [0, kmax ]} 1: Initialisation: z ˆ0|0 ← z 0 , Q0 ← q, Sz = {ˆ z0|0 } 2: for k = 1 to kmax do 3: zˆk|k−1 ←ˆ zk−1|k−1 , Lk ←L, Pk|k−1 ←Pk−1|k−1 +Qk−1 , rk ←min(1, 1.59Lk /ˆ zk|k−1 ), 4: Generate a new seed Rsk and broadcast (Lk , Rsk ) and run Listen-before-Talk protocol 5: Obtain the number of idle slots Nk , and compute yk and vk using (23) and (8) 6: Qk ← q 7: if k ≤ J then 8: φk ← φ 9: else 10: Execute Algorithm 3 11: φk ← output of Algorithm 3 12: end if 13: Calculate Rk and Kk using (27) and (7), and update zˆk|k and Pk|k using (5) and (6) 14: Sz ← Sz ∪ {ˆ zk|k } 15: end for Our main estimation algorithm is illustrated in Algorithm 2. The difference compared to the static scenario is that tag population variation needs to be detected by the CUSUM test presented in Algorithm 3 in the next subsection and the output of Algorithm 3 acts as a feedback to φk , meaning φk is no more a constant after the first J rounds as the static case due to the tag population variation. Specifically, if a change on the tag population is detected in frame k, φk is set to φ to quickly



react to the change, otherwise φk sticks to φ to stabilize the estimation. We note that in the case where zk is constant, Algorithm 2 degenerates to Algorithm 1. C. Detecting Tag Population Change: CUSUM Test The CUSUM Detection Framework. We leverage the CUSUM test to detect the change of tag population and further adjust φk . CUSUM test is a sequential analysis technique typically used for change detection [26]. It is shown to be asymptotically optimal in the sense of the minimum detection time subject to a fixed worst-case expected false alarm rate [27]. In the context of dynamic tag population detection, the reader monitors the innovation process vk = yk − p(ˆ zk|k−1 ). If the number of the tags population is constant, vk equals to uk which is a Gaussian process with zero mean. In contrast, upon the system state changes, i.e., tag population changes, vk drifts away from the zero mean. In our design, we use Φk as a normalised input to the CUSUM test by normalising vk with its estimated standard variance, specified as follows: vk . (29) Φk = q (Pk|k−1 + Qk−1 )Ck 2 + V ar[uk ] zk =ˆ zk|k−1

The reader updates the CUSUM statistics gk+ and gk− as follows: + gk+ = max{0, gk−1 + Φk − Υ}, (30)

g0+ =0

− gk− = min{0, gk−1 + Φk + Υ},

(31)

gk+ = gk− = 0, if δ = 1,

(32)

g0− =

where and 0. And Υ≥0, referred to as reference value, is a filter design parameter indicating the sensitivity of the CUSUM test to the fluctuation of Φk , Moreover, by δ we define an indicator flag indicating tag population change: ( 1 if gk+ > θ or gk− < −θ, δ= (33) 0 otherwise, where θ > 0 is a pre-specified CUSUM threshold. The detailed procedure of the change detection is illustrated in Algorithm 3 where ϕ1 (δ) shown in (37)is used to assign the value to φk according to whether the system state changes. Algorithm 3 CUSUM test: executed by the reader in frame k Input: Υ, θ Output: φk + − 1: Initialisation: g0 ← 0, g0 ← 0 2: Compute Φk using equation (29) + − 3: gk ← (30), gk ← (31) + − 4: if gk > θ or gk < −θ then 5: δ ← 1, φk ← ϕ1 (δ), gk+ ← 0, gk− ← 0 6: else 7: δ ← 0, φk ← ϕ1 (δ) 8: end if 9: Return φk Parameter tuning in CUSUM test. The choice of the threshold θ and the drift parameter Υ has a directly impact on the performance of the CUSUM test in terms of detection

delay and false alarm rate. Formally, the average running length (ARL) L(µ∗ ) is used to denote the duration between two actions [28]. For a large θ, L(µ∗ ) is approximated as ( Θ(θ), if µ∗ 6= 0, (34) L(µ∗ ) = Θ(θ2 ), if µ∗ = 0, where µ∗ denotes the mean of the process Φk . In our context, ARL corresponds to the mean time between two false alarms in the static case and the mean detection delay of the tag population change in the dynamic case. It is easy to see from (34) that a higher value of θ leads to lower false alarm rate at the price of longer detection delay. Therefore, the choices of θ and Υ consists of a tradeoff between the false alarm rate and the detection delay. Recall that Φk can be approximated to a white noise process, i.e, Φk ∼ N [µ∗ , σ ∗ 2 ] with µ∗ = 0, σ ∗ = 1 if the system state does not change. Generically, as recommended in [29], setting θ and Υ as follows achieves good ARL from the engineering perspective. θ = 4σ ∗ , Υ = µ∗ + 0.5σ ∗ .

(35) (36)

In the CUSUM framework, we set φk by ϕ1 (δ) as follows: ( φ, if δ = 1, (37) ϕ1 (δ) = φ, if δ = 0. The rationale is that once a change on the tag population is detected in frame k, φk is set to φ to quickly react to the change, while φk sticks to φ when no system change is detected. Finally, we conclude this section by summarizing the algorithm parameter settings. TABLE I A LGORITHM PARAMETER S ETTINGS Parameters P0|0 , J, q kmax φk Lk , rk θ, Υ

Settings constants as shown in Sec. IX required monitoring time as shown in Sec. IX following Algorithm 3 while satisfying (62) Lk = L, rk = min(1, 1.59L/ˆ zk|k−1 ) with constant L based on (35) and (36) following [29]

VII. P ERFORMANCE A NALYSIS In this section, we establish the stability and the accuracy of our estimation algorithms for both static and dynamic cases. A. Static Case Our analysis is composed of two steps. We first derive the estimation error and then establish the stability and the accuracy of Algorithm 1 in terms of the boundedness of estimation error. Computing Estimation Error. We first approximate the non-linear discrete system by a linear one. To that end, as the function p(zk ) is continuously differentiable at zk = zˆk|k−1 , using the Taylor expansion, we have p(zk ) = p(ˆ zk|k−1 ) + Ck (zk − zˆk|k−1 ) + χ(zk , zˆk|k−1 ), (38)



rk ρ , erk ρ zˆk|k−1 ∞ X 1 rk ρzk j χ(zk , zˆk|k−1 ) = (r ρ − ) . rk ρ j! k e z ˆ k|k−1 j=2

where Ck = −

(39) (40)

Regarding the convergence of χ(zk , zˆk|k−1 ) in (40), assume that zk = a0k zˆk|k−1 , (41) we can obtain the following boundedness of the residual for the case |a0k − 1| < rk1ρ : ∞ X (ˆ zk|k−1 − zk )2 (rk ρ)j+2 zk j − |χ(zk , zˆk|k−1 )| = 1 2 (j + 2)! erk ρ zˆk|k−1 zˆk|k−1 j=0 ≤

(rk ρ)2 (ˆ zk|k−1 − zk )2 , 2 2e(rk ρ) ak zˆk|k−1

(42)

2) The inequality (49) consists of establishing positive Pk|k−1 for every k ≥ 1. 3) As a sufficient condition for stability, the upper bound may be too stringent. As shown in the simulation results, stability is still ensured even with a relatively large . Before the proof of Theorem 1, we first state several auxiliary lemmas to streamline the proof and show how to apply these lemmas to prove Theorem 1 subsequently. Lemma 5. Under the conditions of Theorem 1, if P0|0 > 0, there exist pk , pk > 0 such that the pseudo-covariance Pk|k−1 is bounded for every k ≥ 1, i.e., (51) pk ≤ Pk|k−1 ≤ pk .

Proof: Recall (4) and (6), we have Pk|k−1 ≥ Qk−1 , and Pk|k−1 = Pk−1|k−2 (1 − Kk−1 Ck−1 ) + Qk−1 where ak = 1 − (rk ρ)|1 − (43) 2 2 Pk−1|k−2 Ck−1 Recall the definition of the estimation error in (10) and = P + Qk−1 . (52) k−1|k−2 − Pk−1|k−2 Ck−1 2 + Rk−1 using (20), (3) and (5), we can derive the estimation error Following the design of Rk in (27) and by iteration, we further ek+1|k as follows: get ek+1|k = zk+1 − zˆk+1|k = zk − zˆk|k 1 = zk − zˆk|k−1 − Kk Ck (zk − zˆk|k−1 ) + χ(zk , zˆk|k−1 ) + uk Pk|k−1 = Pk−1|k−2 1 − + Qk−1 = P1|0 1 + φk−1 = (1 − Kk Ck )ek|k−1 + sk + mk , (44) k−2 k−1 Y X k−2 Y 1 1 where sk and mk are defined as · 1− + Qi 1− + Qk−1 . 1 + φi 1 + φj+1 sk = −Kk uk , (45) i=1 i=0 j=i Since φk and Qk are controllable parameters, we can set mk = −Kk χ(zk , zˆk|k−1 ). (46) φk ≤ φ and Qk ≤ q for every k ≥ 0 in Algorithm 1, where Boundedness of Estimation Error. Having derived the φ, q > 0. Consequently, we have dynamics of the estimation error, we now state the main result k−1 j k−1 X 1 1 on the stochastic stability and accuracy of Algorithm 1. 1− Pk|k−1 ≤ P1|0 1 − +q + 1+φ 1+φ j=1 Theorem 1. Consider the discrete-time stochastic system k−1 given by (20) and (25) and Algorithm 1, the estimation error 1 Q ≤ (P + Q ) 1 − + qφ + Qk−1 (53) k−1 0 0|0 ek|k−1 defined by (10) is exponentially bounded in mean 1+φ k−1 square and bounded w.p.o., if the following conditions hold: 1 + qφ + Qk−1 and pk = 1) there are positive numbers q, q, φ and φ such that the Let pk = ((P0|0 + Q0 ) 1 − 1+φ bounds on Qk and φk are satisfied for every k≥0, as in Qk−1 , we have pk ≤ Pk|k−1 ≤ pk . a0k |.

q ≤ Qk ≤ q, φ ≤ φk ≤ φ, 2) The initialization must follow the rules P0|0 > 0, |e1|0 | ≤ with positive real number > 0.

(47) (48) (49) (50)

Remark. By referring to the design objective posed in Section IV, Theorem 1 prove the following properties of our estimation algorithm: • the estimation error of our algorithm is bounded in mean square and the relative estimation error tends to zero; • the estimated population size converges to the real value with exponential rate. Moreover, the conditions in Theorem 1 can be interpreted as follows: 1) The inequalities (47) and (48) can be satisfied by the configuring the correspondent parameters in Algorithm 1, which guarantees the boundedness of the pseudocovariance Pk|k−1 as shown later.

Lemma 6. Let αk , 2

1 1+φk ,

(1 − Kk Ck ) 2 ek|k−1 Pk+1|k

it holds that e2k|k−1 ≤ (1 − αk ) , ∀k ≥ 1. Pk|k−1

(54)

Proof: From (52), we have Pk+1|k = Pk|k−1 (1 − Kk Ck ) + Qk ≥ Pk|k−1 (1 − Kk Ck ) . (55) By substituting it into the left-hand side of (54) and using the fact that Rk = φk Pk|k−1 Ck 2 for every k ≥ 1, we get (1 − Kk Ck )2 (1 − Kk Ck )2 2 ek|k−1 ≤ e2 Pk+1|k Pk|k−1 (1 − Kk Ck ) k|k−1 2 ek|k−1 ek|k−1 2 1 ≤ (1 − Kk Ck ) ≤ 1− . Pk|k−1 1 + φk Pk|k−1 We are thus able to prove (54). Lemma 7. Let bk ,

rk ρ(4ak φk + 1 − ak ) , 2 4ak φk (1 + φk )ˆ zk|k−1 Pk|k−1

it holds

that mk [2(1 − Kk Ck )ek|k−1 + mk ] ≤ bk |ˆ zk|k−1 − zk |3 . Pk+1|k

(56)



Proof: From (46), we get the following expansion mk [2(1 − Kk Ck )ek|k−1 + mk ] Pk+1|k " ! −Pk|k−1 Ck χ(zk , zˆk|k−1 ) Pk|k−1 Ck 2 = · 2 1− Pk+1|k (Pk|k−1 Ck 2 + Rk ) Pk|k−1 Ck 2 + Rk # Pk|k−1 Ck χ(zk , zˆk|k−1 ) . · ek|k−1 − Pk|k−1 Ck 2 + Rk It then follows from (39), (41) and (55) that −Pk|k−1 Ck (rk ρ)2 mk [2(1 − Kk Ck )ek|k−1 + mk ] ≤ r ρ 2 Pk|k−1 Pk+1|k 2e k ak zˆk|k−1 " 2 (ˆ zk|k−1 − zk ) 2Pk|k−1 Ck 2 2 − (1 − Kk Ck )(Pk|k−1 Ck 2 + Rk ) Pk|k−1 Ck 2 + Rk (rk ρ)2 (ˆ zk|k−1 − zk )2 · |ˆ zk|k−1 − zk | − 2 2 1.59 2e ak zˆk|k−1 Pk|k−1 Ck + Rk Pk|k−1 Ck

#

rk ρ(4ak φk + 1 − ak ) ≤ 2 |ˆ zk|k−1 − zk |3 . 4ak φk (1 + φk )ˆ zk|k−1 Pk|k−1 We are thus able to prove (56). 2.46ˆ zk|k−1 sk 2 Lemma 8. E ek|k−1 ≤ when Pk+1|k φk (1 + φk )rk Pk|k−1 rk ρ = 1.59. i h 2 sk = Proof: From (45), we have E Pk+1|k ek|k−1 i h 2 Kk2 E[u2k ] sk e . With (7), (26) and (55), we have E k|k−1 Pk+1|k Pk+1|k ≤

2 e2rk ρ zˆk|k−1 (e−rk ρ −(1+rk ρ)e−2rk ρ ) . 2 φk (1+φkh)Pk|k−1 ρrk i 2 sk Since item E Pk+1|k ek|k−1 influences

the estimation accuracy, we set the optimal persistence probability to minimize 2rk ρ this item. Denote Λ(rk ) = e r2 (e−rk ρ − (1 + rk2 ρ)e−2rk ρ ), k we have dΛ (rk ρ − 2)erk ρ + 2 . = drk rk3 rk ρ

+2) Since rk ρ > 0 and d((rk ρ−2)e = (rk ρ − 1)erk ρ which drk ρ is greater zero if rk ρ > 1 and is smaller than zero if rk ρ < 1, dΛ dΛ and 1) if rk ρ = 1, dr < 0; 2) if rk ρ = 0, dr = 0; 3) if k k dΛ rk ρ = 2, drk > 0, there exists a unique solution rk ρ ∈ (1, 2) dΛ for dr = 0 such that Λ(rk ) is minimized. Searching in (1, 2), k we find the optimal rk ρ = 1.59. Therefore, we can obtain that 2.46ˆ zk|k−1 s2k E ek|k−1 ≤ , ξk , (57) Pk+1|k φk (1 + φk )rk Pk|k−1 which completes the proof. Armed with the above auxiliary lemmas, we next prove Theorem 1. Proof of Theorem 1: First, we construct the following Lyapunov function to define the stochastic process: e2k|k−1 Vk (ek|k−1 ) = , Pk|k−1 which satisfies (4) and (49) as Pk|k−1 > 0. Next, we use Lemma 2 to develop the proof. Because it follows from Lemma 5 that the properties (16) and (17) in Lemma 2 are satisfied, the main task left is to prove (18).

From (44), expanding Vk+1 (ek+1|k ) leads to e2k+1|k [(1 − Kk Ck )ek|k−1 + sk + mk ]2 Vk+1 (ek+1|k ) = = Pk+1|k Pk+1|k 2 mk [2(1 − Kk Ck )ek|k−1 + mk ] (1 − Kk Ck ) 2 = ek|k−1 + Pk+1|k Pk+1|k 2sk [(1 − Kk Ck )ek|k−1 + mk ] s2k + + . Pk+1|k Pk+1|k Furthermore, by Lemmas 6, 7 and 8 and some algebraic operations, we have E Vk+1 (ek+1|k )|ek|k−1 − Vk (ek|k−1 ) ≤ −αk Vk (ek|k−1 ) + bk |ek|k−1 |3 + ξk . (58) To obtain the same formation with (18), we further proceed to bound the second term in bk in (58) as follows: bk |ek|k−1 |3 ≤ ςαk Vk (ek|k−1 ), (59) where 0 < ς < 1 is preset controllable parameter. To prove the above inequality, we need to prove |ek|k−1 | ≤ 4ςa2k φk zˆk|k−1 1.59(4ak φk +1.59|a0k −1|) .

Since |ek|k−1 | = |a0k − 1|ˆ zk|k−1 , it

suffices to show

4ςa2k φk , (60) 1.59(4ak φk + 1.59|a0k − 1|) which is equivalent to (1 − 4φk − 4φk ς)ak 2 + (4φk − 2)ak + 1 ≤ 0 because of (43).√ With some algebraic op|a0k − 1| ≤

erations, we obtain 1) φk < φk >

1 4(1+ς) ; 1 4(1+ς) ;

and 2) and 3)

1−2φk −2 φk (φk +ς) < ak ≤ 1, if 1−4φ√ k (1+ς) 2φk −1+2 φk (φk +ς) ≤ ak ≤ 1, if 4φk (1+ς)−1 1+ς 1 1+2ς ≤ ak ≤ 1, if φk = 4(1+ς) .

√ 2φ −1+2 φk (φk +ς) 1+ς < 1+2ς for every ς Since it holds that k 4φk (1+ς)−1 √ 2φk −1+2 φk (φk +ς) 1 and will decrease monotonically to 1+ς 4φk (1+ς)−1 1 for a large φk , we have in the worst case for φk ≥ 4(1+ς) , 1+ς ≤ ak ≤ 1. (61) 1 + 2ς It follows from the analysis that if we set 1 φk ≥ , (62) 4(1 + ς) (60) can be satisfied. Moreover, it holds that 0.63ς . (63) |a0k − 1| ≤ 1 + 2ς That is, |ek|k−1 | ≤ k , (64)

where k , 0.63ς ˆk|k−1 . By setting φk in (62), for |ek|k−1 | ≤ 1+2ς z k , we thus have E Vk+1 (ek+1|k )|ek|k−1 − Vk (ek|k−1 ) ≤ −(1 − ς)αk Vk (ek|k−1 ) + ξk . (65) Therefore, we are able to apply Lemma 2 to prove Theorem 1 by setting = 0.63ς ˆ1|0 , β ∗ = Q10 , αk∗ = (1 − ς)αk , 1+2ς z 1 βk = p and τk = ξk . k

Remark. Theorem 1 also holds in the sense of Lemma 1 (the off-line version of Lemma 2) by setting the parameters in (15) 1−ς as β = Q10 , α = 1+φ ≤ αk∗ , β = (P0|0 + Q0 + q(φ + 1) ≥ pk , and τ =

Q0 zˆmax φ(1+φ)

≥ ξk , where zˆmax is the maximum estimate.

We conclude the analysis on the performance of our estimation algorithm for the static case with a more profound



investigation on the evolution of the estimation error |ek|k−1 |. More specifically, we can distinguish three regions: q 2.46M zˆk|k−1 • Region 1: φk (M −1)rk (1−ς) ≤ |ek|k−1 | ≤ k . By substituting the condition into the right hand side of (65), we obtain: −(1 − ς)αk Vk (ek|k−1 ) + ξk ≤ k Vk (ek|k−1 ), where M > 1 is a positive con− (1−ς)α M stant and can be set beforehand. It thenfollows that M −(1−ς)αk Vk (ek|k−1 ). E[Vk+1 (ek+1|k )|ek|k−1 ] ≤ M 2 Consequently, we can bound E[ek|k−1 ] as: E[e2k|k−1 ]

(66)

It can then be noted that with = E[e2k|k−1 ] → 0 at an exponential rate as k → ∞. q q 2.46ˆ zk|k−1 2.46M zˆk|k−1 Region 2: φk rk (1−ς) ≤ |ek|k−1 | < φk (M −1)rk (1−ς) . k − (1−ς)α Vk (ek|k−1 ) M

In this case, we have 0. It also follows from Lemma 2 that

E[e2k|k−1 ] ≤

k−1 Y pk (1 − αi∗ ) E[e1|0 2 ] Q0 i=1

+ pk

k−2 X i=1

ξk−i−1

i Y

∗ (1 − αk−j ).

j=1

Hence, when k → ∞, E[e2k|k−1 ] converges exponenPk−2 Qi ∗ tially to pk i=1 ξk−i−1 j=1 (1 − αk−j ) ∼ Θ(ˆ zk|k−1 ), which is decoupled with the initial estimation error and it 1 E[ek|k−1 ] thus holds ≤Θ √ → 0 when zk → ∞. zk zk Combining the above three regions, we get the following results on the convergence of the expected estimation error E[ek|k−1 ]: (1) if the estimation error ispsmall (Region 3), it will converge to a value smaller than Θ( zˆk|k−1 ) as analysed in Region 3; (2) if the estimation error is larger (Region 1), it will decrease as analysed in Region 1 and p fall into either Region 2 or Region 3 where E[ek|k−1 ]≤ Θ( zˆk|k−1 ) such E[ek|k−1 ] that the relative estimation error →0 when zk →∞. zk

(68)

Next, we show the boundedness of the estimation error in Theorem 2. Theorem 2. Under the conditions of Theorem 1, consider the discrete-time stochastic system given by (28) and (25) and Algorithm 2, if there exist time-varying positive real number λk , σk > 0 such that E[wk ] ≤ σk ,

∗ αk−j ).

(67)

which differs from the static case (44) in sk . In the dynamic case, we have

2

Hence, when k → ∞, E[e2k|k−1 ] converges at exponential Pk−2 Qi ∗ rate to pk i=1 ξk−i−1 j=1 (1 − αk−j ) ∼ Θ(ˆ zk|k−1 ), which is decoupled with the initial estimation error and it 1 E[ek|k−1 ] =Θ √ → 0 when zk → ∞. thus holds zk q zk

•

ek+1|k = (1 − Kk Ck )ek|k−1 + sk + mk ,

E[wk ] ≤ λk ,

k−1 Y p ≤ k E[e1|0 2 ] (1 − αi∗ ) Q0 i=1 k−2 X

Our analysis on the stability of Algorithm 2 for the dynamic case is also composed of two steps. First, we derive the estimation error. Second, we establish the stability and the accuracy of Algorithm 2 in terms of the boundedness of estimation error. We first derive the dynamics of the estimation error as follows:

sk = wk − Kk uk

(1−ς)αk . M

αk∗

•

k−1 Y pk 2 ≤ E[e1|0 ] (1 − αi∗ ) Q0 i=1

B. Dynamic Case

(69) (70)

then the estimation error ek|k−1 defined by (10) is exponentially bounded in mean square and bounded w.p.o.. Remark. Note that the condition E[wk ] ≤ λk always holds for E[wk ] < 0, we thus focus on the case that E[wk ] ≥ 0. In the proof, the explicit formulas of λk and σk are derived. As in the static case, the conditions may be too stringent such that the results still hold even if the conditions are not satisfied, as illustrated in the simulations. The proof of Theorem 2 is also based on Lemmas 6, 7 and 8, but due to the introduction of wk into sk , we need another two auxiliary lemmas on E[sk ] and E[s2k ]. Lemma 9. If E[wk ] ≥ 0, then there exists a time-varying real number dk > 0 such that i h 2s [(1 − K C )e k k k k|k−1 + mk ] E ek|k−1 ≤ dk |ek|k−1 |E[wk ]. Pk+1|k Proof: When E[wk ] ≥ 0, from E[vk ] = 0, (41), (55) and the independence between wk and ek|k−1 , we can derive 2sk [(1 − Kk Ck )ek|k−1 + mk ] E ek|k−1 Pk+1|k φk |ek|k−1 | 1.59|ek|k−1 |2 1 + φk ≤ 2E[wk ] + φk Pk|k−1 1 + φk 2ak (1 + φk )ˆ zk|k−1 2ak φk + (1 − ak ) ≤ E[wk ] |ek|k−1 |. ak φk Pk|k−1 By setting dk =

2ak φk + (1 − ak ) , ak φk Pk|k−1

(71)

we thus complete the proof. Lemma 10.h There exists ai time-varying parameter ξk∗ > 0 sk 2 such that E Pk+1|k ek|k−1 ≤ ξk∗ .


This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TCOMM.2016.2592524, IEEE Transactions on Communications 10 2

Proof: By (68), we have s2k = wk2 − 2Kk wk uk + Kk2 u2k . Since wk and uk are uncorrelated and ek|k−1 does not depend on either wk or uk , we have E[wk2 ] K 2 E[u2k ] s2k ek|k−1 = + k . (72) E Pk+1|k Pk+1|k Pk+1|k Substituting (7), (55) and Lemma i 8, noticing 2 that h using (1+φ )E[w ] s2k E[uk ] = 0, we get E Pk+1|k ek|k−1 ≤ φk Pkk|k−1 k + 2.46ˆ zk|k−1 φk (1+φk )rk Pk|k−1 .

E[wk 2 ] ≤

Thirdly, let 2

4.92ˆ zk|k−1 φk (M − 1)rk (1 − ς)|ek|k−1 | ≥ , 2 2M (1 + φk ) (1 + φk )2

Finally, by setting ξk∗ as

s

9.84M zˆk|k−1 , ˜, φk (M − 1)rk (1 − ς) 2.46ˆ zk|k−1 , σk . E[wk 2 ] ≤ (1 + φk )2

|ek|k−1 | ≥

Furthermore, bounding the second item in bk as (59) and given φk in (62), yields E Vk+1 (ek+1|k )|ek|k−1 − Vk (ek|k−1 ) ≤ −(1 − ς)αk Vk (ek|k−1 ) + dk |ek|k−1 |E[wk ] + ξk∗

k−1 Y p ≤ k E[e1|0 2 ] (1 − αi∗ ). Q0 i=1

Secondly, substituting (71), (73) into (75) leads to ak φk (1 − ς)|ek|k−1 | E[wk ] ≤ , (76) (1 + φk ) (2ak φk + 1 − ak )

(80)

However, since a0k and ak are unknown a priori, we thus need to transform the right hand side of (76) to ς a computable form. From (61), we get a1k − 1 ≤ 1+ς such that it holds for the right hand side of (76) that ak φk (1−ς)|ek|k−1 | φk (1−ς)˜ 3(1+φk )(2ak φk +1−ak ) ≥ 3(1+φk )(2φk + ς ) . 1+ς Finally, let E[wk ] ≤

•

φk (1 − ς)˜ 3(1 + φk ) 2φk +

ς 1+ς

, λk ,

(81)

we can establish (74) and thus get that E[e2k|k−1 ] → 0 at an exponential q rate when k → ∞. q 9.84ˆ zk|k−1 9.84M zˆk|k−1 Region 2: φk rk (1−ς) ≤ |ek|k−1 | < φk (M −1)rk (1−ς) . Given ˜, λk and σk as in Region 1, in this case, we have −(1 − ς)αk Vk (ek|k−1 ) + dk |ek|k−1 |E[wk ] + ξk∗ ≤ 0. It then follows from Lemma 2 that E[e2k|k−1 ] ≤

k−1 Y pk E[e1|0 2 ] (1 − αi∗ ) Q0 i=1

+ pk

(74)

That is, it should hold that dk |ek|k−1 |E[wk ] + ξk∗ ≤ M −1 M (1 − ς)αk Vk (ek|k−1 ). To that end, we firstly let the following inequalities hold ( −1 dk |ek|k−1 |E[wk ] ≤ M2M (1 − ς)αk Vk (ek|k−1 ), (75) M −1 ∗ ξk ≤ 2M (1 − ς)αk Vk (ek|k−1 ).

(79)

The rational behind can be interpreted as follows: i) the right term of (77) cannot be less than zero and ii) there always exists the measurement uncertainty in the system. Consequently, the impact of tag population change plus the measurement uncertainty should equal in order of magnitude that of only measurement uncertainty, which can be achieved by establishing E[wk 2 ] ≤ Kk2 E[uk 2 ] and (78) with reference to (72) and (73).

≤ −αk Vk (ek|k−1 ) + bk |ek|k−1 |3 + dk |ek|k−1 |E[wk ] + ξk∗ .

E[e2k|k−1 ]

(78)

and we thus have

2.46ˆ zk|k−1 1 + φk E[wk 2 ] + , (73) φk Pk|k−1 φk (1 + φk )rk Pk|k−1 we complete the proof. Armed with the above lemmas, we next prove Theorem 2 by utilizing the same method with the proof of Theorem 1. Proof of Theorem 2: Recall (45) and (68), we notice that the only difference between the estimation errors of Algorithms 2 and 1 is sk . Therefore, it suffices to study the impact of wk on Vk (ek|k−1 ). It follows from Lemmas 6, 7, 8, 9 and 10 that E Vk+1 (ek+1|k )|ek|k−1 − Vk (ek|k−1 ) ξk∗ =

for |ek|k−1 | ≤ k . ˆ1|0 , And we can thus prove Theorem 2 by setting = 0.63ς 1+2ς z ∗ β = Q10 , αk∗ = (1 − ς)αk , τk = ξk∗ + dk |ek|k−1 |λk and βk = p1 . k We conclude the analysis on the performance of our estimation algorithm for the dynamic case with a more profound investigation on the evolution of the estimation error |ek|k−1 | and derive the explicit formulas for λk and σk . More specifically, we can distinguish three regions: q 9.84M zˆk|k−1 • Region 1: φk (M −1)rk (1−ς) ≤ |ek|k−1 | ≤ k . In this case, the objective is to achieve E Vk+1 (ek+1|k )|ek|k−1 − Vk (ek|k−1 ) 1 ≤ − (1 − ς)αk Vk (ek|k−1 ) M so that E[e2k|k−1 ] is bounded as

φk (M − 1)rk (1 − ς)|ek|k−1 | 2M (1 + φk )2 2.46ˆ zk|k−1 − . (77) (1 + φk )2

k−2 X i=1

τk−i−1

i Y

∗ (1 − αk−j ).

j=1

Hence, when k → ∞, E[e2k|k−1 ] converges exponentially Pk−2 Qi ∗ to pk i=1 τk−i−1 j=1 (1 − αk−j ) ∼ Θ(ˆ zk|k−1 ) and •

E[ek|k−1 ] zk

= Θ( √1zk ) → 0 for zk → ∞. q 9.84ˆ zk|k−1 Region 3: 0 ≤ |ek|k−1 | < φk rk (1−ς) . The circumstances in this region are very complicated due to E[wk ] and E[wk 2 ], we here thus just consider the worst case that E[wk ] = λk and E[wk 2 ] = σk . Consequently, we have −(1 − ς)αk Vk (ek|k−1 ) + dk |ek|k−1 |E[wk ] + ξk∗ > 0, it thus holds that



and it then follows from Lemma 2 that E[e2k|k−1 ] ≤

IX. N UMERICAL A NALYSIS

k−1 Y pk E[e1|0 2 ] (1 − αi∗ ) Q0 i=1

+ pk

k−2 X

τk−i−1

i=1

i Y

∗ (1 − αk−j ).

j=1

Hence, when k → ∞, E[e2k|k−1 ] converges at exponential Pk−2 Qi ∗ ) ∼ Θ(ˆ zk|k−1 ), rate to pk i=1 τk−i−1 j=1 (1 − αk−j and thus

E[ek|k−1 ] zk

≤ Θ( √1zk ) → 0 for zk → ∞.

Note that for the case that E[wk ] < λk and E[wk 2 ] < σk , the range of Region 3 will shrink and the range of Region 2 will largen. Integrating the above three regions, we can get the similar results on the convergence of the expected estimation error E[ek|k−1 ] as in the static case. VIII. D ISCUSSION This section discusses the application of the proposed algorithm to the unreliable channel and multi-reader scenario. Error-prone channel. The unreliable channel may corrupt an idle slot into a busy slot and vice versa. We consider the random error model as [30]. Let t0 and t1 (0 ≤ t1 , t2 < 0.5) be the false positive rate that an empty slot turns into a busy slot and the false negative rate, respectively. For notation convenience, we mark parameters in the perfect channel model with a superscript ∗ to define their counterparts under the imperfect channel model. With some straightforward algebraic operations, we have p∗ (Zk ) = t1 + (1 − t0 − t1 )p(Zk ), ∗

2

V ar [uk ] = (1 − t0 − t1 ) V ar[uk ].

(82) (83)

We can compute the new Kalman gain Kk∗ as Kk∗

1 Kk . = (1 − t0 − t1 )

In this section, we conduct extensive simulations to evaluate the performance of the proposed tag population estimation algorithms by focusing on the relative estimation error denoted z −ˆz as REEk = k zk|k−1 . Specifically, unless otherwise speck ified, we simulate in sequence both static and dynamic RFID systems where the initial tag population are z0 = 104 with the following default parameters: q = 0.1, P0|0 = 1, J = 3, θ = 4 and Υ = 0.5 with reference to (35) and (36), L = 1500, φ = 0.25 and φ = 100 such that (62) always holds. Since the proposed algorithms do not require collision detection, we set a slot to 0.4ms as in the EPCglobal C1G2 standard [12]. We will discuss the effect of φ and φ on the performance in next section. A. Algorithm Verification In the subsection, we show the impact of φ and φ on the system performance. To that end, with REE0 =0.5, we keep zk =104 in static scenario p while the tag population varies in order of magnitude from zˆk|k−1 to 0.4ˆ zk|k−1 in different patterns in dynamic scenario. Specifically, we set φ=100 while varying φ=0.25, 0.5, 1 in Fig. 1, 2, and fix φ=0.25 with varying φ=1, 10, 100 in Fig. 3, 4. As shown in the figures, a smaller φ leads to rapider convergence rate while the bigger φ, the smaller the deviation. Thus, we choose φ=0.25 and φ=100 in the rest of the simulation. Moreover, we make the following observations. First, as derived in Theorem 2, the estimation is stable and accurate facing to a relative small p population change, i.e., around the order of magnitude zˆk|k−1 . Second, the proposed scheme also functions nicely even when the estimation error is as high as 0.4ˆ zk|k−1 tags as shown in Fig. 2 and 4. This is due to the CUSUM-based change detection which detects state changes promptly such that a small value is set for φk , leading to rapid convergence rate. B. Algorithm Performance

(84)

The ideal channel case is equivalent to the case where t0 = t1 = 0. It is then straightforward to derive the stability in the case of imperfect channel by using p∗ (Zk ), V ar∗ [uk ] and Kk∗ . In this regard, we find that Theorem 1 and 2 still hold under the same conditions, meaning that our approach is robust against channel errors. Multi-reader case. In multi-reader scenarios, we leverage the same approach as [31]. The main idea is that a backend server can be used to synchronize all readers such that the RFID system with multiple readers operates as the singlereader case. Specially, the back-end server calculates all the parameters and sends them to all readers such that they broadcast the same parameters to the tags. Subsequently, each reader sends its bitmap to the back-end server. Then the back-end server applies OR operator on all bitmaps, which eliminates the impact of the duplicate readings of tags in the overlapped interrogation region.

In this section, we evaluate the performance of the proposed EKF-based estimator, referred to as EEKF here, in comparison with [7] in static scenario and with [19] in dynamic scenario. 1) Static System (zk = 104 ): We evaluate the performance by varying initial relative error as z0 −ˆ z0|0 • REE0 = =0.2 implies a small initial estimation z0 error and satisfies (64) with 0.5≤ς< 1. • REE0 = 0.5 means a medium initial estimation error. • REE0 = 0.8 means a large initial estimation error. In this three case, we simulate EZB with the optimal parameters as specified in [7] with the accuracy 0.95, specifically, when REE0 =0.2, frame size L = 35, the number of trails n = 70 and persistence probability p0 = 0.0057; when REE0 =0.5, then L=66, n=43 and p0 =0.0113; when REE0 =0.8, then L=215, n=20 and p0 =0.0449. For legible presentation, we set the simulation time here to 215 ∗ 30 based on the case REE0 =0.8, then kmax = 185, 97, 30, respectively. Note that we use the same frame size with each case of EZB.



1.2

0.5 0

5

25

Static:φ=100. x 10

Fig. 2.

0.8 0

1.5

50

100

EEKF

0.15 0.1 0.05 50

100 Time (frame)

150

(a) REE0 = 0.2

x 10

25

Actual φ=1 φ=10 φ=100

0.94 0.92 0.9 0

5

Fig. 3.

10 15 Time (frame)

20

40

60

80

0.2 0.15 0.1 0.05 20

40 60 Time (frame)

(b) REE0 = 0.5

80

0.8

Actual φ=1 φ=10 φ=100 5

Fig. 4. 1.5

EEKF

1

0.5 0

25

Static:φ=0.25.

1

0 0

20

4

× 10

10 15 Time (frame)

20

25

Dynamic:φ=0.25.

4

EZB

EEKF

1.0 0.5 0 0.2

5

10

5

10

15

20

25

30

15 20 Time (frame)

25

30

0.15 0.1 0.05 0 0

(c) REE0 = 0.8

Algorithm performance under different initial estimation errors.

Fig. 5 illustrates the estimation processes with different initial estimation errors. As shown in the figures, the estimation zˆk|k−1 converges towards the actual number of tags within very short time in all cases, despite the initial estimation error. Though EZB is faster than EEKF to stabilize around the actual size with the error less than 0.05, EEKF achieves smaller deviation. TABLE II E XECUTION TIME Algorithm JREP EEKF

0.96

EZB

0.5 0

150

0.2

0 0

20

Dynamic:φ=100.

1

Fig. 5.

10 15 Time (frame)

4

EZB

No. of tags

5

No. of tags

1.2

20

0.5 0

Estimation error

Fig. 1.

10 15 Time (frame)

Actual φ=0.25 φ=0.5 φ=1

Number of tags(×104)

0.6

0.8

0.98

No. of tags

Actual φ=0.25 φ=0.5 φ=1

0.7

1

Estimation error

0.8

Estimation error


0.9

1.2

1

4

Number of tags(×10 )


1

10→12.5 3.2768 1.2

Variation of tag population (×103 ) →6.737 →9.364 →7.049 →8.616 3.2768 3.2768 3.2768 3.2768 1.2 1.2 0.6 0.6

2) Dynamic system: In this subsection, we evaluate the performance of EEKF for dynamic systems by comparing with the start-of-the-art solution JREP [19] in terms of execution time to achieve the required accuracy. To that end, we refer to the simulation setting in [19]. Specifically, the initial estimation error is 10%. The tag population size changes by following the normal distribution with the mean of 10000 and the variance of 20002 and the accuracy requirement is 95%. By taking 5 samplings, we obtain the results as listed in Table II. As shown in Table II, EEKF is more time-efficient than JREP. This is because the persistence probability in JREP is set to optimise the power-of-two frame size, which increases the variance of the number of empty slots and leads to the performance degradation. In contrast, EEKF can minimise this variance while promptly detecting the tag population changes. X. C ONCLUSION In this paper, we have addressed the problem of tag estimation in dynamic RFID systems and designed a generic framework of stable and accurate tag population estimation

schemes based on Kalman filter. Technically, we leveraged the techniques in extended Kalman filter (EKF) and cumulative sum control chart (CUSUM) to estimate tag population for both static and dynamic systems. By employing Lyapunov drift analysis, we mathematically characterised the performance of the proposed framework in terms of estimation accuracy and convergence speed by deriving the closed-form conditions on the design parameters under which our scheme can stabilise around the real population size with bounded relative estimation error that tends to zero within exponential convergence rate. R EFERENCES [1] RFID Journal. DoD releases final RFID policy. [Online]. [2] RFID Journal. DoD reaffirms its RFID goals. [Online]. [3] Chun-Hee Lee and Chin-Wan Chung. Efficient storage scheme and query processing for supply chain management using RFID. In ACM SIGMOD, pages 291–302. ACM, 2008. [4] Lionel M Ni, Dian Zhang, and Michael R Souryal. RFID-based localization and tracking technologies. IEEE Wireless Communications, 18(2):45–51, 2011. [5] Po Yang, Wenyan Wu, Mansour Moniri, and Claude C Chibelushi. Efficient object localization using sparsely distributed passive RFID tags. IEEE Trans. on Industrial Electronics, 60(12):5914–5924, 2013. [6] RFID Journal. Wal-Mart begins RFID process changes. [Online]. [7] Murali Kodialam, Thyaga Nandagopal, and Wing Cheong Lau. Anonymous tracking using RFID tags. In IEEE INFOCOM, pages 1217–1225. IEEE, 2007. [8] Tao Li, Samuel Wu, Shigang Chen, and Mark Yang. Energy efficient algorithms for the RFID estimation problem. In IEEE INFOCOM, pages 1–9. IEEE, 2010. [9] Chen Qian, Hoilun Ngan, Yunhao Liu, and Lionel M Ni. Cardinality estimation for large-scale RFID systems. IEEE Trans. on Parallel and Distributed Systems, 22(9):1441–1454, 2011. [10] Muhammad Shahzad and Alex X Liu. Every bit counts: fast and scalable RFID estimation. In ACM Mobicom, pages 365–376, 2012. [11] Yuanqing Zheng and Mo Li. Zoe: Fast cardinality estimation for largescale RFID systems. In IEEE INFOCOM, pages 908–916. IEEE, 2013. [12] EPCglobal Inc. Radio-frequency identity protocols class-1 generation-2 UHF RFID protocol for communications at 860 mhz - 960 mhz version 1.0.9. [Online].



[13] Yongkyu Song and Jessy W Grizzle. The extended Kalman filter as a local asymptotic observer for nonlinear discrete-time systems. In American Control Conference, pages 3365–3369. IEEE, 1992. [14] Murali Kodialam and Thyaga Nandagopal. Fast and reliable estimation schemes in RFID systems. In ACM Mobicom, pages 322–333. ACM, 2006. [15] Hao Han, Bo Sheng, Chiu Chiang Tan, Qun Li, Weizhen Mao, and Sanglu Lu. Counting RFID tags efficiently and anonymously. In IEEE INFOCOM, pages 1–9. IEEE, 2010. [16] Venkatesh Sarangan, MR Devarapalli, and Sridhar Radhakrishnan. A framework for fast RFID tag reading in static and mobile environments. Computer Networks, 52(5):1058–1073, 2008. [17] Lei Xie, Bo Sheng, Chiu Chiang Tan, Hao Han, Qun Li, and Daoxu Chen. Efficient tag identification in mobile RFID systems. In IEEE INFOCOM, pages 1–9. IEEE, 2010. [18] Qingjun Xiao, Bin Xiao, and Shigang Chen. Differential estimation in dynamic RFID systems. In IEEE INFOCOM, pages 295–299. IEEE, 2013. [19] Qingjun Xiao and Min Chen Shigang Chen Yian Zhou. Temporally or spatially dispersed joint rfid estimation using snapshots of variable lengths. In ACM MobiHoc, pages 247–256. ACM, 2015. [20] Toader Morozan. Boundedness properties for stochastic systems. In Stability of Stochastic Dynamical Systems, pages 21–34. Springer, 1972. [21] Tzyh-Jong Tarn and Yona Rasis. Observers for nonlinear stochastic systems. IEEE Trans. Automatic Control, 21(4):441–448, 1976. [22] Konrad Reif, Stefan G¨unther, E Yaz Sr, and Rolf Unbehauen. Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans. on Automatic Control, 44(4):714–728, 1999. [23] Matthew B Rhudy and Yu Gu. Online stochastic convergence analysis of the Kalman filter. International Journal of Stochastic Analysis, 2013, 2013. [24] Klaus Finkenzelle. RFID handbook: Radio frequency identification fundamentals and applications. John Wiley & Sons, 2000. [25] V F Kolchin, B A Sevastyanov, and V P Chistyakov. Random allocation. Wiley New York, 1978. [26] Fredrik Gustafsson and Fredrik Gustafsson. Adaptive filtering and change detection. Wiley New York, 2000. [27] E Brodsky and Boris S Darkhovsky. Nonparametric methods in change point problems. Springer Science & Business Media, 1993. [28] Mich`ele Basseville, Igor V Nikiforov, et al. Detection of abrupt changes: theory and application. Prentice Hall Englewood Cliffs, 1993. [29] Fred Spiring. Introduction to statistical quality control. Technometrics, 49(1):108–109, 2007. [30] Mei Chen, Wan Luo, Zhen Mo, Shigang Chen, and Yi Fang. An efficient tag search protocol in large-scale RFID systems with noisy channel. IEEE/ACM TON, 2015. [31] Muhammad Shahzad and Alex X Liu. Expecting the unexpected: Fast and reliable detection of missing RFID tags in the wild. In IEEE INFOCOM, pages 1939–1947, 2015.

Jihong Yu received the B.E degree in communication engineering and M.E degree in communication and information systems from Chongqing University of Posts and Telecommunications, Chongqing, China, in 2010 and 2013, respectively, and is currently pursuing the Ph.D. degree in computer science at the University of Paris-Sud, Orsay, France. His research interests include wireless communications and networking and RFID technologies.

Lin Chen (S07-M10) received his B.E. degree in Radio Engineering from Southeast University, China in 2002 and the Engineer Diploma from Telecom ParisTech, Paris in 2005. He also holds a M.S. degree of Networking from the University of Paris 6. He currently works as associate professor in the department of computer science of the University of Paris-Sud. He serves as Chair of IEEE Special Interest Group on Green and Sustainable Networking and Computing with Cognition and Cooperation, IEEE Technical Committee on Green Communications and Computing. His main research interests include modeling and control for wireless networks, distributed algorithm design and game theory.

Rongrong Zhang received the B.Eng and M.Eng degrees in communication and information systems from Chongqing University of Posts and Telecommunications, Chongqing, China, in 2010 and 2013, respectively, and is currently pursuing the Ph.D. degree in Computer Communication in the Faculty of Mathematics and Computer Science at the University of Paris Descartes, France. Her research interests focus on Wireless Area Body Networks (WBANs) for healthcare and wireless communications.

Kehao Wang received the B.S degree in Electrical Engineering, M.s degree in Communication and Information System from Wuhan University of Technology, Wuhan, China, in 2003 and 2006, respectively, and Ph.D in the Department of Computer Science, the University of Paris-Sud XI, Orsay, France, in 2012. He currently works as associate professor in the department of Information Engineering of Wuhan University of Technology. His research interests are cognitive radio networks, wireless network resource management, and data hiding.


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